68
68

Sep 18, 2013
09/13

by
Volker Runde

texts

######
eye 68

######
favorite 0

######
comment 0

We give a short biographical sketch of Karl Weierstrass.

Source: http://arxiv.org/abs/math/0202308v1

1
1.0

Jun 29, 2018
06/18

by
Vincent Bouchard; Nitin K. Chidambaram; Tyler Dauphinee

texts

######
eye 1

######
favorite 0

######
comment 0

We study the connection between the Eynard-Orantin topological recursion and quantum curves for the family of genus one spectral curves given by the Weierstrass equation. We construct quantizations of the spectral curve that annihilate the perturbative and non-perturbative wave-functions. In particular, for the non-perturbative wave-function, we prove, up to order hbar^5, that the quantum curve satisfies the properties expected from matrix models. As a side result, we obtain an infinite...

Topics: High Energy Physics - Theory, Algebraic Geometry, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1610.00225

39
39

Sep 21, 2013
09/13

by
Josef Schicho; David Sevilla

texts

######
eye 39

######
favorite 0

######
comment 0

We define the concept of Tschirnhaus-Weierstrass curve, named after the Weierstrass form of an elliptic curve and Tschirnhaus transformations. Every pointed curve has a Tschirnhaus-Weierstrass form, and this representation is unique up to a scaling of variables. This is useful for computing isomorphisms between curves.

Source: http://arxiv.org/abs/0808.3038v2

34
34

Sep 23, 2013
09/13

by
Zoe Laing; David Singerman

texts

######
eye 34

######
favorite 0

######
comment 0

We look for Riemann surfaces whose automorphism group acts transitively on the Weierstrass points. We concentrate on hyperelliptic surfaces, surfaces with PSL(2, q) as automorphism group, Platonic surfaces and Fermat curves.

Source: http://arxiv.org/abs/1012.2020v1

39
39

Sep 23, 2013
09/13

by
RJ Cova; WJ Zakrzewski

texts

######
eye 39

######
favorite 0

######
comment 0

We present our results of a numerical investigation of the behaviour of a system of two solitons in the (2+1) dimensional $CP^1$ model on a torus. Defined by the elliptic function of Weierstrass, and working in the Skyrme version of the model, the soliton lumps exhibit splitting, scattering at right angles and motion reversal in the various configurations considered. The work is restricted to systems with no initial velocity.

Source: http://arxiv.org/abs/hep-th/0109007v1

42
42

Sep 23, 2013
09/13

by
Guy Laville; Ivan Ramadanoff

texts

######
eye 42

######
favorite 0

######
comment 0

It will be shown that the Stone-Weierstrass theorem for Clifford-valued functions is true for the case of even dimension. It remains valid for the odd dimension if we add a stability condition by principal automorphism.

Source: http://arxiv.org/abs/math/0411090v2

11
11

Jun 26, 2018
06/18

by
T. Shaska; C. Shor

texts

######
eye 11

######
favorite 0

######
comment 0

In this lecture we give a brief introduction to Weierstrass points of curves and computational aspects of $q$-Weierstrass points on superelliptic curves.

Topics: Mathematics, Algebraic Geometry, Complex Variables

Source: http://arxiv.org/abs/1502.06285

50
50

Jul 20, 2013
07/13

by
Alberto Saa; Roberto Venegeroles

texts

######
eye 50

######
favorite 0

######
comment 0

The Weierstrass random walk is a paradigmatic Markov chain giving rise to a L\'evy-type superdiffusive behavior. It is well known that Special Relativity prevents the arbitrarily high velocities necessary to establish a superdiffusive behavior in any process occurring in Minkowski spacetime, implying, in particular, that any relativistic Markov chain describing spacetime phenomena must be essentially Gaussian. Here, we introduce a simple relativistic extension of the Weierstrass random walk and...

Source: http://arxiv.org/abs/1007.3314v2

356
356

Dec 28, 2013
12/13

by
Patrick Bruskiewich

texts

######
eye 356

######
favorite 0

######
comment 0

To simplify an integral that is a rational function in cos(x) or sin(x), a substitution of the form t = tan(ax/2) will convert the integrand into an ordinary rational function in t. This substitution, is known as the Weierstrass Substitution, and honours the mathematician, Karl Weierstrass (1815-1897) who developed the technique.

Topics: Weierstrass Substitution, Integration, Calculus, mathematics, mathematician, cosine, sine, rational...

54
54

Sep 23, 2013
09/13

by
Nathan Kaplan; Lynnelle Ye

texts

######
eye 54

######
favorite 0

######
comment 0

We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitz' necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. We also show that the family of semigroups known to be Weierstrass semigroups using a result of Eisenbud and Harris, has zero density in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup.

Source: http://arxiv.org/abs/1202.6331v2

40
40

Sep 19, 2013
09/13

by
Jaume Giné; Maite Grau

texts

######
eye 40

######
favorite 0

######
comment 0

The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define {\it Weierstrass integrability} and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of integrable systems there are non--Liouvillian integrable systems.

Source: http://arxiv.org/abs/0909.0113v1

0
0.0

Jun 29, 2018
06/18

by
Nathan Pflueger

texts

######
eye 0

######
favorite 0

######
comment 0

We define a class of numerical semigroups S, which we call Castelnuovo semigroups, and study the subvariety $M^S_{g,1}$ of $M_{g,1}$ consisting of marked smooth curves with Weierstrass semigroup S. We determine the number of irreducible components of these loci and determine their dimensions. Curves with these Weierstrass semigroups are always Castelnuovo curves, which provides the basic tool for our argument. This analysis provides examples of numerical semigroups for which $M^S_{g,1}$ is...

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1608.08178

0
0.0

Jun 30, 2018
06/18

by
Jiryo Komeda; Takeshi Takahashi

texts

######
eye 0

######
favorite 0

######
comment 0

We investigate the number of Galois Weierstrass points whose Weierstrass semigroups are generated by two positive integers.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1703.09416

40
40

Sep 23, 2013
09/13

by
David Harbater; Julia Hartmann; Daniel Krashen

texts

######
eye 40

######
favorite 0

######
comment 0

We prove a form of the Weierstrass Preparation Theorem for normal algebraic curves over complete discrete valuation rings. While the more traditional algebraic form of Weierstrass Preparation applies just to the projective line over a base, our version allows more general curves. This result is then used to obtain applications concerning the values of u-invariants, and on the period-index problem for division algebras, over fraction fields of complete two-dimensional rings. Our approach uses...

Source: http://arxiv.org/abs/1109.6362v2

71
71

Sep 23, 2013
09/13

by
Silviu Teleman

texts

######
eye 71

######
favorite 0

######
comment 0

This paper extends a version of the Stone-Weierstrass theorem to more general C*-algebras. Namely, assume that A is a unital, not necessarily separable, C*-algebra, and B is a C*-subalgebra containing the unit element. Then, I prove that: If B separates the factorial states of A, then B=A. This generalizes a result of Popa and Longo for the case when A is separable. A true Stone-Weierstrass theorem would state that, if B separates the pure states of A, then B=A. This problem is open even in the...

Source: http://arxiv.org/abs/1111.2980v2

2
2.0

Jun 27, 2018
06/18

by
Florence Fauquant-Millet; Anthony Joseph

texts

######
eye 2

######
favorite 0

######
comment 0

Adapted pairs and Weierstrass sections are central to the invariant theory associated to the action of an algebraic Lie algebra a on a finite dimensional vector space X. In this a need not be a semisimple Lie algebra. Here their general properties are described particularly when a is the canonical truncation of a biparabolic subalgebra of a simple Lie algebra and X is the dual of a.

Topics: Representation Theory, Mathematics

Source: http://arxiv.org/abs/1503.02523

31
31

Sep 19, 2013
09/13

by
Caterina Cumino; Eduardo Esteves; Letterio Gatto

texts

######
eye 31

######
favorite 0

######
comment 0

Let C be the union of two general connected, smooth, nonrational curves X and Y intersecting transversally at a point P. Assume that P is a general point of X or of Y. Our main result, in a simplified way, says: Let Q be a point of X. Then Q is the limit of special Weierstrass points on a family of smooth curves degenerating to C if and only if Q is not P and either of the following conditions hold: Q is a special ramification point of the linear system |K_X+(g_Y+1)P|, or Q is a ramification...

Source: http://arxiv.org/abs/math/0701663v1

25
25

Sep 19, 2013
09/13

by
Rainer Fuhrmann; Fernando Torres

texts

######
eye 25

######
favorite 0

######
comment 0

We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves.

Source: http://arxiv.org/abs/alg-geom/9709013v1

5
5.0

Jun 28, 2018
06/18

by
Alain J. Brizard

texts

######
eye 5

######
favorite 0

######
comment 0

A consistent notation for the Weierstrass elliptic function $\wp(z;g_{2},g_{3})$, for $g_{2} > 0$ and arbitrary values of $g_{3}$ and $\Delta \equiv g_{2}^{3} - 27 g_{3}^{2}$, is introduced based on the parametric solution for the motion of a particle in a cubic potential. These notes provide a roadmap for the use of {\sf Mathematica} to calculate the half-periods $(\omega_{1},\omega_{3},\omega_{2} \equiv \omega_{1} + \omega_{3})$ of the Weierstrass elliptic function.

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1510.07818

30
30

Sep 17, 2013
09/13

by
L. Martina; Kur. Myrzakul; R. Myrzakulov

texts

######
eye 30

######
favorite 0

######
comment 0

The study of the relation between the Weierstrass inducing formulae for constant mean curvature surfaces and the completely integrable euclidean nonlinear sigma-model suggests a connection among integrable sigma -models in a background and other type of surfaces. We show how a generalization of the Weierstrass representation can be achieved and we establish a connection with the Weingarten surfaces. We suggest also a possible generalization for two-dimensional surfaces immersed in a flat space...

Source: http://arxiv.org/abs/math/0207261v1

25
25

texts

######
eye 25

######
favorite 0

######
comment 0

0
0.0

Jun 29, 2018
06/18

by
Nathan Pflueger

texts

######
eye 0

######
favorite 0

######
comment 0

We give an upper bound on the codimension in $M_{g,1}$ of the variety $M^S_{g,1}$ of marked curves $(C,p)$ with a given Weierstrass semigroup. The bound is a combinatorial quantity which we call the effective weight of the semigroup; it is a refinement of the weight of the semigroup, and differs from it precisely when the semigroup is not primitive. We prove that whenever the effective weight is less than g, the variety $M^S_{g,1}$ is nonempty and has a component of the predicted codimension....

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1608.05666

35
35

Sep 19, 2013
09/13

by
Arnaldo Garcia; Fernando Torres

texts

######
eye 35

######
favorite 0

######
comment 0

We study geometrical properties of maximal curves having classical Weierstrass gaps.

Source: http://arxiv.org/abs/math/9801106v1

47
47

Sep 18, 2013
09/13

by
Fernando Torres

texts

######
eye 47

######
favorite 0

######
comment 0

We prove that the constellation of Weierstrass points characterizes the isomorphism-class of double covering of curves of genus large enough.

Source: http://arxiv.org/abs/alg-geom/9604006v1

42
42

Sep 18, 2013
09/13

by
Grzegorz Banaszak; Jan Milewski

texts

######
eye 42

######
favorite 0

######
comment 0

In this paper we introduce new definition of Hodge structures and show that $\R$-Hodge structures are determined by $\R$-linear operators that are annihilated by the Weierstrass $\sigma$-function

Source: http://arxiv.org/abs/1209.0806v1

77
77

Nov 14, 2013
11/13

by
Katrin Wehrheim

texts

######
eye 77

######
favorite 0

######
comment 0

Until Weierstrass published his shocking paper in 1872, most of the mathematical world (including luminaries like Gauss) believed that a continuous function could only fail to be differentiable at some collection of isolated points. In fact, it turns out that �most� continuous functions are non-differentiable at all points. (To understand what this statement could mean, you should take courses in topology and measure theory.) However, Weierstrass was not, in fact, the �rst to construct...

Topics: Maths, Analysis and Calculus, Mathematics

Source: http://www.flooved.com/reader/1197

36
36

Sep 20, 2013
09/13

by
Yu. V. Brezhnev

texts

######
eye 36

######
favorite 0

######
comment 0

We present some new results in theory of classical theta-functions of Jacobi and sigma-functions of Weierstrass: ordinary differential equations (dynamical systems) and series expansions. The paper is basically organized as a stream of new formulas.

Source: http://arxiv.org/abs/math/0601371v3

36
36

Sep 21, 2013
09/13

by
Enrico Arbarello; Gabriele Mondello

texts

######
eye 36

######
favorite 0

######
comment 0

We show that the locally closed strata of the Weierstrass flags on the moduli spaces of curves of genus g and on the moduli space of curves of genus g with one marked point are almost never affine.

Source: http://arxiv.org/abs/1204.6669v1

6
6.0

Jun 26, 2018
06/18

by
Sheng Zhang; Brendan Harding

texts

######
eye 6

######
favorite 0

######
comment 0

We established a new method called Discrete Weierstrass Fourier Transform, a faster and more generalized Discrete Fourier Transform, to approximate discrete data. The theory of this method as well as some experiments are analyzed in this paper. In some examples, this method has a faster convergent speed than Discrete Fourier Transform.

Topics: Mathematics, Numerical Analysis

Source: http://arxiv.org/abs/1502.07734

34
34

Sep 23, 2013
09/13

by
Robin de Jong

texts

######
eye 34

######
favorite 0

######
comment 0

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation of the hyperelliptic discriminant of X/S, and the valuation of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable these invariants are known to satisfy certain inequalities. We prove an exact formula relating the two invariants with intersection...

Source: http://arxiv.org/abs/0710.5464v2

5
5.0

Jun 30, 2018
06/18

by
Claudia Alfes; Michael Griffin; Ken Ono; Larry Rolen

texts

######
eye 5

######
favorite 0

######
comment 0

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We show that mock modular forms which arise from Weierstrass $\zeta$-functions encode the central $L$-values and $L$-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by H\"ovel, we...

Topics: Mathematics, Number Theory

Source: http://arxiv.org/abs/1406.0443

68
68

Sep 22, 2013
09/13

by
P. Bracken; P. P. Goldstein; A. M. Grundland

texts

######
eye 68

######
favorite 0

######
comment 0

The connection between the complex Sine and Sinh-Gordon equations on the complex plane associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlev\'e test and analyse the possibility of deriving the B\"acklund transformation from the singularity analysis of the complex Sine-Gordon equation. We make use of the analysis using the known relations for the Painlev\'{e} equations to...

Source: http://arxiv.org/abs/nlin/0111058v1

0
0.0

Jun 30, 2018
06/18

by
Gareth Jones; Jonathan Kirby; Tamara Servi

texts

######
eye 0

######
favorite 0

######
comment 0

We explain which Weierstrass elliptic functions are locally definable from other elliptic functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way.

Topics: Mathematics, Logic

Source: http://arxiv.org/abs/1402.3963

113
113

Sep 18, 2013
09/13

by
Dilcia Perez; Yamilet Quintana

texts

######
eye 113

######
favorite 0

######
comment 0

The celebrated and famous Weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials. This theorem is the first significant result in Approximation Theory of one real variable and plays a key role in the development of General Approximation Theory. Our aim is to investigate some new results relative to such theorem, to present a history of the subject, and to introduce some open problems.

Source: http://arxiv.org/abs/math/0611038v2

3
3.0

Jun 29, 2018
06/18

by
Alexander Polishchuk

texts

######
eye 3

######
favorite 0

######
comment 0

We construct an open substack $U\subset\mathcal{M}_{g,1}$ with the complement of codimension $\ge 2$ and a morphism from $U$ to a weighted projective stack, which sends the Weierstrass locus $\mathcal{W}\cap U$ to a point, and maps $\mathcal{M}_{g,1}\setminus\mathcal{W}$ isomorphically to its image. The proof uses alternative birational models of $\mathcal{M}_{g,1}$ and $\mathcal{M}_{g,2}$ from arXiv:1509.07241.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1611.04243

32
32

Sep 23, 2013
09/13

by
Jinbi Jin

texts

######
eye 32

######
favorite 0

######
comment 0

Starting from the classical division polynomials we construct homogeneous polynomials a_n, b_n, c_n such that for P = (x:y:z) on an elliptic curve in Weierstrass form over an arbitrary ring we have nP = (a_n(P):b_n(P):c_n(P)). To show that a_n, b_n, c_n indeed have this property we use the a priori existence of such polynomials, which we deduce from the Theorem of the Cube.

Source: http://arxiv.org/abs/1303.4327v1

78
78

Jul 19, 2013
07/13

by
Sotiris Karanikolopoulos; Aristides Kontogeorgis

texts

######
eye 78

######
favorite 0

######
comment 0

The relation of the Weierstrass semigroup with several invariants of a curve is studied. For Galois covers of curves with group $G$ we introduce a new filtration of the group decomposition subgroup of $G$. The relation to the ramification filtration is investigated in the case of cyclic covers. We relate our results to invariants defined by Boseck and we study the one point ramification case. We also give applications to Hasse-Witt invariant and symmetric semigroups.

Source: http://arxiv.org/abs/1005.2871v1

1
1.0

Jun 30, 2018
06/18

by
Francois Greer

texts

######
eye 1

######
favorite 0

######
comment 0

Let $X\to \mathbb P^2$ be the elliptic Calabi-Yau threefold given by a general Weierstrass equation. We answer the enumerative question of how many discrete rational curves lie over lines in the base, proving part of a conjecture by Huang, Katz, and Klemm. The key inputs are a modularity theorem of Kudla and Millson for locally symmetric spaces of orthogonal type and the deformation theory of $A_n$ singularities.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1701.06596

31
31

Sep 21, 2013
09/13

by
C. Arezzo; G. P. Pirola; M. Solci

texts

######
eye 31

######
favorite 0

######
comment 0

In this note we prove a Weierstrass representation formula for pluriminimal submanifolds of euclidean spaces. We use this formula to produce new families of examples of pluriminimal submanifolds. We also prove that any affine algebraic manifold can be pluriminimally embedded into some euclidean space in a non holomorphic manner.

Source: http://arxiv.org/abs/math/0104124v1

43
43

Sep 18, 2013
09/13

by
Shigeki Matsutani

texts

######
eye 43

######
favorite 0

######
comment 0

This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dirac operator, gives the data of the immersion. In the derivation, the simple Frobenius reciprocity of Clifford algebras $S$ and $M$ plays important roles.

Source: http://arxiv.org/abs/math/0605255v4

23
23

Sep 17, 2013
09/13

by
Kenneth Kunen

texts

######
eye 23

######
favorite 0

######
comment 0

C(X) denotes the space of continuous complex-valued functions on the compact Hausdorff space X. X has the CSWP if every subalgebra of C(X) which separates points and contains the constant functions is dense in C(X). W. Rudin showed that all scattered X have the CSWP. We describe a class of non-scattered X with the CSWP; by another result of Rudin, such X cannot be metrizable.

Source: http://arxiv.org/abs/math/0305076v2

0
0.0

Jun 29, 2018
06/18

by
L. Beshaj

texts

######
eye 0

######
favorite 0

######
comment 0

We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers $\mathcal O_{\mathbb F}$. This is done via reduction theory and Julia invariant of binary sextics. We show that when the binary sextics has extra automorphisms this is usually easier to compute. Moreover, we show that when the curve is given in the standard form $y^2=f(x^2)$, where $f(x)$ is a monic polynomial, $f(0)=1$ which is defined over $\mathcal O_{\mathbb F}$ then this form is reduced.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1612.08318

0
0.0

Jun 30, 2018
06/18

by
M. Irene Falcão; Fernando Miranda; Ricardo Severino; M. Joana Soares

texts

######
eye 0

######
favorite 0

######
comment 0

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas which motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper we...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1702.04935

35
35

Sep 18, 2013
09/13

by
Grzegorz Banaszak; Jan Milewski

texts

######
eye 35

######
favorite 0

######
comment 0

A $\sigma$-operator on a complexification $V_{\C}$ of an $\R$-vector space $V_{\R}$ is an operator $A \in \rm{End}_{\C} (V_{\C})$ such that $\sigma (A) = 0$ where $\sigma (z)$ denotes the Weierstrass $\sigma$-function. In this paper we define the notion of the strongly pseudo-real $\sigma$-operator and prove that there is one to one correspondence between real mixed Hodge structures and strongly pseudo-real $\sigma$-operators.

Source: http://arxiv.org/abs/1211.0687v1

41
41

Sep 19, 2013
09/13

by
Abel Castorena

texts

######
eye 41

######
favorite 0

######
comment 0

Let C be a hyperelliptic Riemann surface. We show that the hyperelliptic Weierstrass points of C are non-degenerated critical points of Morse index +2 of the curvature function K of the Theta metric on C (called also Bergman metric). When the genus of C is two, we compute all critical points of K and we show in this case that K is a Morse function.

Source: http://arxiv.org/abs/math/0703075v2

76
76

Nov 14, 2013
11/13

by
Katrin Wehrheim

texts

######
eye 76

######
favorite 1

######
comment 0

All known examples of non-differentiable continuous functions are constructed in a similar fashion to the following example � they are limits of functions that oscillate more and more on small scales, but with higher-frequency oscillations being damped quickly. The example we give here is a faithful reproduction of Weierstrass�s original 1872 proof. It is somewhat more complicated than the example given as Theorem 7.18 in Rudin, but is superior in at least one important way, as explained in...

Topics: Maths, Analysis and Calculus, Mathematics

Source: http://www.flooved.com/reader/1276

54
54

Sep 18, 2013
09/13

by
Dmitry Korotkin; Vasilisa Shramchenko

texts

######
eye 54

######
favorite 0

######
comment 0

The goal of this paper is to propose a new way to generalize the Weierstrass sigma-function to higher genus Riemann surfaces. Our definition of the odd higher genus sigma-function is based on a generalization of the classical representation of the elliptic sigma-function via Jacobi theta-function. Namely, the odd higher genus sigma-function $\sigma_{\chi}(u)$ (for $u\in \C^g$) is defined as a product of the theta-function with odd half-integer characteristic $\beta^{\chi}$, associated with a...

Source: http://arxiv.org/abs/1201.3961v1

73
73

Sep 23, 2013
09/13

by
Jiryo Komeda; Shigeki Matsutani; Emma Previato

texts

######
eye 73

######
favorite 0

######
comment 0

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular form have pointed to the relevance of $\tau$-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) $\sigma$-function. This paper proposes a construction of $\sigma$-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface as a generalization...

Source: http://arxiv.org/abs/1303.0451v2

0
0.0

Jun 30, 2018
06/18

by
Elena Yu. Bunkova

texts

######
eye 0

######
favorite 0

######
comment 0

We consider the coefficients in the series expansion at zero of the Weierstrass sigma function \[ \sigma(z) = z \sum_{i, j \geqslant 0} {a_{i,j} \over (4 i + 6 j + 1)!} \left({g_2 z^4 \over 2}\right)^i \left(2 g_3 z^6\right)^j. \] We have $a_{i,j} \in \mathbb{Z}$. We present the divisibility Hypothesis for the integers $a_{i,j}$ \begin{align*} \nu_2(a_{i,j}) &= \nu_2((4i + 6j + 1)!) - \nu_2(i!) - \nu_2(j!) - 3 i - 4 j, & \nu_3(a_{i,j}) &= \nu_3((4i + 6j + 1)!) - \nu_3(i!) -...

Topics: Algebraic Topology, Complex Variables, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1701.00848

32
32

Sep 22, 2013
09/13

by
Stefan C. Mancas; Greg Spradlin; Harihar Khanal

texts

######
eye 32

######
favorite 0

######
comment 0

In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified BBM equation. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant make the equation integrable in terms of...

Source: http://arxiv.org/abs/1302.5408v2